Units are the superheroes of number fields, swooping in to save the day with their multiplicative inverses. They form a group that's key to understanding the structure of rings of integers and solving tricky Diophantine equations.
is the ultimate cheat code for units. It tells us exactly how many fundamental units we need to generate the whole group, based on the field's embeddings. This powerful tool unlocks deeper insights into number fields.
Units in Number Fields
Defining Units and Their Properties
Top images from around the web for Defining Units and Their Properties
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
Root of unity modulo n - Wikipedia, the free encyclopedia View original
Is this image relevant?
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 3
Top images from around the web for Defining Units and Their Properties
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
Root of unity modulo n - Wikipedia, the free encyclopedia View original
Is this image relevant?
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
Dirichlet eta function - Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 3
Units in a number field K are elements of the ring of integers OK with multiplicative inverses within OK
Form a multiplicative group called the of K
of a unit always equals ±1 due to multiplicative property of norms
Algebraic integers whose minimal polynomial has a constant term of ±1
Set of units forms a finitely generated abelian group (Dirichlet's unit theorem)
Play crucial role in unique factorization of ideals in OK
Torsion subgroup consists of all roots of unity in the number field
Importance and Applications of Units
Enable factorization of ideals in OK
Help determine of number fields
Used in solving Diophantine equations
Crucial in studying arithmetic properties of number fields
Aid in constructing integral bases for number fields
Provide insights into the structure of rings of integers
Applications in cryptography (elliptic curve cryptosystems)
Structure of Unit Groups
Dirichlet's Unit Theorem
Unit group isomorphic to direct product of finite cyclic group and
Rank of free abelian part equals r+s−1
r: number of real embeddings
s: number of pairs of complex embeddings
Finite cyclic part consists of roots of unity in the number field
Fundamental units generate entire unit group with roots of unity
measures "size" of fundamental units
Important in class number formulas
Real quadratic fields have unit group of rank 1
Imaginary quadratic fields have finite unit group (only roots of unity)
Examples of Unit Group Structures
Q(2): Unit group {±1}×⟨1+2⟩
Q(−1): Unit group {±1,±i}
Q(ζ3): Unit group {±1,±ζ3,±ζ32}
Cyclotomic fields Q(ζn): Unit group generated by cyclotomic units
Totally real cubic fields: Unit group of rank 2
CM fields: Unit group structure related to that of its maximal real subfield
Finite Generation of Unit Groups
Logarithmic Embedding Approach
Proof relies on logarithmic embedding of unit group into real vector space
Define logarithmic map from unit group to Rr+s
Maps unit to logarithms of absolute values of its embeddings
Image contained in hyperplane of dimension r+s−1 (product formula)
Kernel of map is finite (roots of unity in the number field)
Dirichlet's approximation theorem shows image forms lattice in hyperplane
Conclude finite generation by combining finite kernel and discrete lattice image
Key Steps in the Proof
Construct logarithmic map L:U→Rr+s
Show Im(L) lies in hyperplane H:x1+⋯+xr+2xr+1+⋯+2xr+s=0
Prove ker(L) is finite using properties of algebraic numbers
Apply Dirichlet's approximation theorem to find units close to any point in H
Use geometry of numbers to show Im(L) is a lattice in H
Combine results to prove unit group is finitely generated
Unit Groups: Examples
Quadratic Fields
Real quadratic fields Q(d), d>0
Find fundamental unit using continued fraction expansion of d
Example: Q(5) has fundamental unit 21+5
Imaginary quadratic fields Q(d), d<0
Q(−1): Unit group {±1,±i}
Q(−2): Unit group {±1}
Q(−3): Unit group {±1,±21+−3,±2−1+−3}
Higher Degree Fields
Cubic fields: Use norm equations and algebraic manipulations
Verify units by checking norms and generation of known small-height units
Cyclotomic fields Q(ζn): Describe unit group using cyclotomic units
Example: Q(ζ5) has unit group generated by ζ5 and ζ52−1ζ5−1
Use regulator and class number formula to verify completeness of fundamental units
Example: For real cubic field, compute regulator R and verify R≈2πh∣D∣ where h is class number and D is discriminant
Key Terms to Review (18)
Additive unit: An additive unit is an element in a ring that, when added to another element, does not change the value of that element. In algebraic structures, additive units play a crucial role in defining the properties of the unit group, which includes understanding how units interact under addition and how they contribute to the overall structure of the group.
Class number: The class number is an important invariant in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It reflects how many distinct ideal classes exist, indicating whether every ideal can be expressed uniquely as a product of prime ideals. A class number of one means that unique factorization holds, while a higher class number suggests complications in the structure of ideals within the number field.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for modern mathematics and significantly influenced the development of algebraic number theory.
Dirichlet's Unit Theorem: Dirichlet's Unit Theorem is a fundamental result in algebraic number theory that describes the structure of the group of units in the ring of integers of a number field. It states that the unit group is isomorphic to a finite direct product of two components: a finite torsion subgroup and a free abelian group whose rank is given by the number of real embeddings minus the number of complex embeddings of the number field.
Free abelian group: A free abelian group is a type of algebraic structure that consists of a set equipped with an operation that satisfies the group axioms, where every element can be uniquely expressed as a finite sum of basis elements multiplied by integers. This structure allows for the elements to be added together and multiplied by integers without any relations other than those required by the group properties, making it fundamentally important for understanding units and their interactions in number theory.
Hermann Minkowski: Hermann Minkowski was a prominent mathematician known for his contributions to number theory and geometry, particularly in relation to the study of algebraic integers and the theory of units. His work laid the groundwork for understanding the structure of unit groups in number fields and introduced important bounds that are critical for determining class numbers, which measure the failure of unique factorization in rings of integers.
Ideal Class Group: The ideal class group is a fundamental concept in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It consists of equivalence classes of fractional ideals, where two ideals are considered equivalent if their product with a principal ideal is again a fractional ideal. This group plays a crucial role in understanding the structure of rings of integers and their relationship to number fields, helping to connect various areas such as discriminants, integral bases, and the properties of Dedekind domains.
Multiplicative unit: A multiplicative unit is an element in a ring that has a multiplicative inverse within that ring, meaning when multiplied by its inverse, the product is the identity element of multiplication, which is 1. This concept is crucial in understanding the structure of the unit group, where all multiplicative units form a group under multiplication, showcasing properties such as closure and the existence of inverses.
Norm: In algebraic number theory, the norm of an algebraic number is a value that gives important information about its behavior in relation to a field extension. It can be viewed as a multiplicative measure that reflects how the number scales when considered within its minimal field, connecting properties of elements with their corresponding fields and extensions.
Regulator: The regulator is a crucial concept in algebraic number theory that measures the size of the unit group of a number field. It captures the logarithmic growth of units and is fundamentally linked to the structure of the unit group, specifically through its role in the connection between units and class numbers. Understanding regulators helps to unveil the intricacies of Dirichlet's unit theorem, which describes the relationship between units in number fields and their ranks.
Torsion Units: Torsion units are elements of the unit group of a ring or number field that have finite order, meaning they become equal to one after being raised to some power. In the context of algebraic number theory, torsion units represent a crucial part of understanding the structure of the unit group, as they highlight the periodic nature of certain units within the group.
Unit Group: The unit group of a ring is the set of elements that have multiplicative inverses within that ring. Understanding unit groups is crucial for exploring the structure of algebraic objects, particularly in relation to the behavior of integers and their generalizations, which often manifest in the study of rings and fields.
Unit lattice: A unit lattice is a mathematical structure that represents the set of units in a ring of integers in a number field. It provides a way to visualize the relationships between units, showing how they can be combined through addition and multiplication to form new units. This concept is essential for understanding the structure of the unit group, which consists of all invertible elements in the ring and plays a crucial role in algebraic number theory.
Unit rank: Unit rank is defined as the number of independent units in the unit group of a ring or an algebraic number field. It provides insight into the structure of the unit group, which consists of all elements in a ring that have multiplicative inverses. Understanding unit rank helps in analyzing how many free generators can exist, and it is tied to important concepts such as the structure theorem for finitely generated abelian groups.
Unit Theorem for Real Quadratic Fields: The Unit Theorem for Real Quadratic Fields states that the unit group of the ring of integers in a real quadratic field is finitely generated and has a specific structure involving roots of unity and a free part. This theorem highlights the behavior of units in these fields, revealing that the unit group can be decomposed into a torsion subgroup, which consists of roots of unity, and a free abelian group whose rank is determined by the field's discriminant. Understanding this structure is crucial for further studies in algebraic number theory.
Units in \( \mathbb{Q}(\sqrt{d}) \): Units in \( \mathbb{Q}(\sqrt{d}) \) are the elements of the ring of integers of the quadratic field \( \mathbb{Q}(\sqrt{d}) \) that have a multiplicative inverse also within that ring. These units are significant as they form a group under multiplication, reflecting the structure of the unit group and providing insights into the algebraic properties of the field. Understanding units helps in exploring Diophantine equations and classifying quadratic fields based on their unit groups.
Units in z[i]: Units in $\mathbb{Z}[i]$, the ring of Gaussian integers, are elements that have a multiplicative inverse also within the ring. Specifically, these units are the numbers that can be multiplied by another element in $\mathbb{Z}[i]$ to yield 1, the multiplicative identity. The existence of these units is key to understanding the structure of the unit group in this context, as it helps identify the elements that maintain the property of invertibility within the ring.
Valuation: A valuation is a function that assigns a non-negative real number to elements of a number field, providing a way to measure the size or 'distance' of those elements from zero. This concept is crucial as it connects various aspects of number theory, including understanding the structure of number fields, the behavior of units, the principles of approximation, and the process of completing number fields. Valuations help in determining properties such as divisibility and congruences within these mathematical constructs.